In algebraic geometry, the normal cone
of a subscheme
of a scheme
is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.
Definition
The normal cone or
of an embedding , defined by some sheaf of ideals ''I'' is defined as the
relative Spec
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
When the embedding ''i'' is
regular the normal cone is the normal bundle, the vector bundle on ''X'' corresponding to the dual of the sheaf .
If ''X'' is a point, then the normal cone and the normal bundle to it are also called the
tangent cone
In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities.
Definitions in nonlinear analysis
In nonlinear analysis, there are many definitions for a tangen ...
and the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
(
Zariski tangent space
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, an ...
) to the point. When ''Y'' = Spec ''R'' is affine, the definition means that the normal cone to ''X'' = Spec ''R''/''I'' is the Spec of the
associated graded ring In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring:
:\operatorname_I R = \oplus_^\infty I^n/I^.
Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the gra ...
of ''R'' with respect to ''I''.
If ''Y'' is the product ''X'' × ''X'' and the embedding ''i'' is the
diagonal embedding In algebraic geometry, given a morphism of schemes p: X \to S, the diagonal morphism
:\delta: X \to X \times_S X
is a morphism determined by the universal property of the fiber product X \times_S X of ''p'' and ''p'' applied to the identity 1_X : X ...
, then the normal bundle to ''X'' in ''Y'' is the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
to ''X''.
The normal cone (or rather its projective cousin) appears as a result of blow-up. Precisely, let
be the blow-up of ''Y'' along ''X''. Then, by definition, the exceptional divisor is the pre-image
; which is the
projective cone of
. Thus,
The global sections of the normal bundle classify
embedded infinitesimal deformations of ''Y'' in ''X''; there is a natural bijection between the set of closed subschemes of , flat over the ring ''D'' of dual numbers and having ''X'' as the special fiber, and ''H''
0(''X'', ''N''
''X'' ''Y'').
Properties
Compositions of regular embeddings
If
are
regular embeddings, then
is a regular embedding and there is a natural exact sequence of vector bundles on ''X'':
If
are regular embeddings of codimensions
and if
is a regular embedding of codimension
then
In particular, if
is a
smooth morphism In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if
*(i) it is locally of finite presentation
*(ii) it is flat, and
*(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular.
(iii) mea ...
, then the normal bundle to the
diagonal embedding In algebraic geometry, given a morphism of schemes p: X \to S, the diagonal morphism
:\delta: X \to X \times_S X
is a morphism determined by the universal property of the fiber product X \times_S X of ''p'' and ''p'' applied to the identity 1_X : X ...
(''r''-fold) is the direct sum of copies of the
relative tangent bundle .
If
is a closed immersion and if
is a flat morphism such that
, then
If
is a
smooth morphism In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if
*(i) it is locally of finite presentation
*(ii) it is flat, and
*(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular.
(iii) mea ...
and
is a regular embedding, then there is a natural exact sequence of vector bundles on ''X'':
(which is a special case of an exact sequence for
cotangent sheaves.)
Cartesian square
For a Cartesian square of schemes
with
the vertical map, there is a closed embedding
of normal cones.
Dimension of components
Let
be a scheme of finite type over a field and
a closed subscheme. If
is of ; i.e., every irreducible component has dimension ''r'', then
is also of pure dimension ''r''. (This can be seen as a consequence of
#Deformation to the normal cone.) This property is a key to an application in intersection theory: given a pair of closed subschemes
in some ambient space, while the
scheme-theoretic intersection In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y.
Locally, ...
has irreducible components of various dimensions, depending delicately on the positions of
, the normal cone to
is of pure dimension.
Examples
Let
be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is
Non-regular Embedding
Consider the non-regular embedding
then, we can compute the normal cone by first observing
If we make the auxiliary variables
and
we get the relation
We can use this to give a presentation of the normal cone as the relative spectrum
Since
is affine, we can just write out the relative spectrum as the affine scheme
giving us the normal cone.
Geometry of this normal cone
The normal cone's geometry can be further explored by looking at the fibers for various closed points of
. Note that geometrically
is the union of the
-plane
with the
-axis
,
so the points of interest are smooth points on the plane, smooth points on the axis, and the point on their intersection. Any smooth point on the plane is given by a map
for
and either
or
. Since it's arbitrary which point we take, for convenience let's assume
, hence the fiber of
at the point
is isomorphic to